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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the honeycomb conjecture for a class of minimal convex partitions


Authors: Dorin Bucur, Ilaria Fragalà, Bozhidar Velichkov and Gianmaria Verzini
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 52C20, 51M16, 65N25, 49Q10
DOI: https://doi.org/10.1090/tran/7326
Published electronically: May 17, 2018
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Abstract: We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every $ n \in \mathbb{N}$ the minimizers of $ F$ among convex $ n$-gons with given area. In particular, our result allows us to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover, we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of $ \lambda _1$ among convex pentagons, hexagons, and heptagons with prescribed area.


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Additional Information

Dorin Bucur
Affiliation: Institut Universitaire de France Laboratoire de Mathématiques UMR 5127 Université de Savoie, Campus Scientifique 73376 Le-Bourget-Du-Lac, France
Email: dorin.bucur@univ-savoie.fr

Ilaria Fragalà
Affiliation: Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano, Italy
Email: ilaria.fragala@polimi.it

Bozhidar Velichkov
Affiliation: Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38401 Saint-Martin-d’Hères, France
Email: bozhidar.velichkov@univ-grenoble-alpes.fr

Gianmaria Verzini
Affiliation: Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci, 32 20133 Milano, Italy
Email: gianmaria.verzini@polimi.it

DOI: https://doi.org/10.1090/tran/7326
Keywords: Optimal partitions, honeycomb conjecture, Cheeger constant, logarithmic capacity, discrete Faber-Krahn inequality.
Received by editor(s): March 10, 2017
Published electronically: May 17, 2018
Additional Notes: This work was partially supported by the PRIN-2015KB9WPT Grant: “Variational methods, with applications to problems in mathematical physics and geometry”, by the ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”, and by the INDAM-GNAMPA group.
Article copyright: © Copyright 2018 American Mathematical Society

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